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PY3P05

o Rotational transitions

o Vibrational transitions

o Electronic transitions

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o Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated.

o This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj):

o Involves the following assumptions:

o Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed.

o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons.

!

"molecule (ˆ r i, ˆ R j ) ="electrons( ˆ r i, ˆ R j )"nuclei( ˆ R j )

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o Electronic transitions: UV-visible

o Vibrational transitions: IR

o Rotational transitions: Radio

Electronic Vibrational Rotational

E

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o Must first consider molecular moment of inertia:

o At right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”.

o Generally specified about three axes: Ia, Ib, Ic.

o For linear molecules, the moment of inertia about the internuclear axis is zero.

o See Physical Chemistry by Atkins.

!

I = miri 2

i "

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o Rotation of molecules are considered to be rigid rotors.

o Rigid rotors can be classified into four types:

o Spherical rotors: have equal moments of inertia (e.g., CH4, SF6).

o Symmetric rotors: have two equal moments of inertial (e.g., NH3).

o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).

o Asymmetric rotors: have three different moments of inertia (e.g., H2O).

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o The classical expression for the energy of a rotating body is:

where !a is the angular velocity in radians/sec.

o For rotation about three axes:

o In terms of angular momentum (J = I!):

o We know from QM that AM is quantized:

o Therefore, , J = 0, 1, 2, …

!

Ea =1/2Ia"a 2

!

E =1/2Ia"a 2 +1/2Ib"b

2 +1/2Ic"c 2

!

E = Ja 2

2Ia + Jb 2

2Ib + Jc 2

2Ic

!

J = J(J +1)!2

!

EJ = J(J +1)! 2I

, J = 0, 1, 2, …

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o Last equation gives a ladder of energy levels.

o Normally expressed in terms of the rotational constant, which is defined by:

o Therefore, in terms of a rotational term:

cm-1

o The separation between adjacent levels is therefore

F(J) - F(J-1) = 2BJ

o As B decreases with increasing I =>large molecules have closely spaced energy levels.

!

hcB = ! 2

2I => B = !

4"cI

!

F(J) = BJ(J +1)

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o Transitions are only allowed according to selection rule for angular momentum:

"J = ±1

o Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor.

o Note, the intensity of each line reflects the populations of the initial level in each case.

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o Consider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement

(F = -kx). Potential energy is therefore

V = 1/2 kx2

o Can write the corresponding Schrodinger equation as

where

o The SE results in allowed energies

!

!2

2µ d2" dx 2

+ [E #V ]" = 0

!2

2µ d2" dx 2

+ [E #1/2kx 2]" = 0

!

µ = m1m2 m1 + m2

!

Ev = (v +1/2)!"

!

" = k µ

#

$ % &

' (

1/ 2

v = 0, 1, 2, …

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o The vibrational terms of a molecule can therefore be given by

o Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond.

o A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k.

!

G(v) = (v +1/2) ˜ v

!

˜ v = 1 2"c

k µ

#

$ % &

' (

1/ 2

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o The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations.

o Transition occur for "v = ±1

o This potential does not apply to energies close to dissociation energy.

o In fact, parabolic potential does not allow molecular dissociation.

o Therefore more consider anharmonic oscillator.

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o A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations.

o At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit.

o Must therefore use a asymmetric potential. E.g., The Morse potential:

where De is the depth of the potential minimum and

!

V = hcDe 1" e "a(R"Re )( )

2

!

a = µ" 2

2hcDe

#

$ %

&

' (

1/ 2

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o The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels:

where xe is the anharmonicity constant:

o The second term in the expression for G increases with v => levels converge at high quantum numbers.

o The number of vibrational levels for a Morse oscillator is finite:

v = 0, 1, 2, …, vmax

!

G(v) = (v +1/2) ˜ v " ( ˜ v +1/2)2 xe ˜ v

!

xe = a2! 2µ"

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o Molecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J)

o Selection rules obtained by combining rotational selection rule !J = ±1 with vibrational rule !v = ±1.

o When vibrational transitions of the form v + 1 ! v occurs, !J = ±1.

o Transitions with !J = -1 are called the P branch:

o Transitions with !J = +1 are called the R branch:

o Q branch are all transitions with !J = 0

!

S(v,J) = (v +1/2) ˜ v + BJ(J +1)

!

˜ v P (J) = S(v +1,J "1) " S(v,J) = ˜ v " 2BJ

!

˜ v R (J) = S(v +1,J +1) " S(v,J) = ˜ v + 2B(J +1)

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o Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV).

o Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy).

P branch

Q branch

R branch

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o Electronic transitions occur between molecular orbitals.

o Must adhere to angular momentum selection rules.

o Molecular orbitals are labeled, ", #, $, … (analogous to S, P, D, … for atoms)

o For atoms, L = 0 => S, L = 1 => P o For molecules, % = 0 => ", % = 1 => #

o Selection rules are thus

$% = 0, ±1, $S = 0, $"=0, $& = 0, ±1

o Where & = % + " is the total angular momentum (orbit and spin).

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